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\author{王立庆（2022级跨专业选修课班级）}
\title{复变函数教学大纲（学生使用）}
\date{2024年3月8日}

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\begin{document}

\maketitle

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\section*{时间地点}
\begin{enumerate}
\itemsep0em 
\item 上课时间地点：周一下午16:45 - 18:20（9-10节），六教409.
\item 答疑时间地点：周一晚上20:30 - 21:30, 周二下午13:00 - 17:00, 一教210. 
\end{enumerate}

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\section*{使用教材}
\begin{enumerate}\itemsep0em 
\item 钟玉泉，复变函数论，高等教育出版社，2021年3月第五版。
\end{enumerate}

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\section*{参考文献}
\begin{enumerate}\itemsep0em 
\item  Lars V. Ahlfors, Complex Analysis, China Machine Press, February 2022. 
\item  阿尔福斯著，赵志勇、薛运华、杨旭译，复分析，机械工业出版社，2023年6月第一版。


\end{enumerate}

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\section*{课程成绩}
\begin{enumerate}
\item  平时成绩 100\%.%包括课堂考勤、课外作业、期中考试、阶段测验。
\begin{enumerate}\itemsep0em 
\item[1.1.] 课堂考勤10次，共20分。
\item[1.2.] 课外作业10次，共30分。
\item[1.3.] 期中考试1次，共20分。
\item[1.4.] 期末考试1次，共30分。
\end{enumerate}

%\item  期末成绩 60\%, 计算题和应用题。

\end{enumerate}

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\section*{主要内容}
\begin{enumerate}\itemsep0em 
\item  第一章：复数的运算，复变函数的极限与连续。
\item  第二章：解析函数，柯西-黎曼方程，指数函数，三角函数，双曲函数，根式函数，对数函数。
\item  第三章：复变函数的积分，柯西积分定理，柯西积分公式，刘维尔定理，调和函数。
\item  第四章：复数项幂级数，解析函数的泰勒展式，解析函数的零点的孤立性，最大模原理。
\item  第五章：解析函数的洛朗展式，孤立奇点的三种类型，整函数与亚纯函数。
\item  第六章：留数定理，函数在无穷远点的留数，用留数定理计算实积分，幅角原理。
\item  *第七章：解析变换的保角性，初等函数所构成的共形映射，黎曼存在与唯一性定理。
\item  *第八章：解析延拓的概念，幂级数延拓，黎曼-施瓦茨对称原理，单值性定理，黎曼面的概念。
\item  *第九章：调和函数，平均值定理，极值原理，泊松积分公式，狄利克雷问题。

\end{enumerate}

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%\section*{重要定理}
%\begin{table}[ht!]\centering
%\begin{tabular}{|p{1cm}|p{12cm}|}  \hline 
% 章节 & 冠名定理 \\ \hline 
% 1.3 & 伯恩斯坦定理 \\ \hline

%\end{tabular}
%\end{table}

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\newpage
\section*{授课计划}

\begin{table}[ht!]\centering
\begin{tabular}
{|p{1cm}|p{1cm}|p{3.5cm}|p{1cm}|p{7cm}|}  \hline 
周 & 章 & 内容 & 节 &内容  \\ \hline \hline
1  & 1 & 复数与复变函数  	& 1.1 & 复数     \\  \hline
    &	&				& 1.2 & 复平面上的点集     \\  \hline 
2  &	&				 & 1.3 & 复变函数     \\  \hline
    &	&				 & 1.4 & 复球面与无穷远点     \\  \hline \hline
3  & 2 & 解析函数 		& 2.1 & 解析函数的概念与柯西-黎曼方程     \\  \hline
    &	&				 & 2.2 & 初等解析函数     \\  \hline 
4  &	&				 & 2.3 & 初等多值函数     \\  \hline \hline
5  & 3 & 复变函数的积分    & 3.1 & 复积分的概念及其简单性质     \\  \hline
    &	&				 & 3.2 & 柯西积分定理     \\  \hline
6  &	&				 & 3.3 & 柯西积分公式及其推论     \\  \hline
    &	&				 & 3.4 & 解析函数与调和函数的关系     \\  \hline \hline
 7  & 4 & 解析函数的幂级数   & 4.1 & 复级数的基本性质    \\  \hline
    &	&				 & 4.2 & 幂级数     \\  \hline
8  &	&				 & 4.3 & 解析函数的泰勒展式     \\  \hline
    &	&				 & 4.4 & 解析函数的零点的孤立性及唯一性定理     \\  \hline \hline
9  & & 		& & {\color{red}期中考试}  \\  \hline \hline
10 & 5 & 解析函数的洛朗展式     & 5.1 & 解析函数的洛朗展式     \\  \hline
     &	&				 & 5.2 & 解析函数的孤立奇点     \\  \hline
11 &	&				 & 5.3 & 解析函数在无穷远点的性质     \\  \hline
     &	&				 & 5.4 & 整函数与亚纯函数的概念     \\  \hline \hline
12 & 6 & 留数理论及其应用     & 6.1 & 留数     \\  \hline
     &	&				 & 6.2 & 用留数定理计算实积分    \\  \hline
13 &	&				 & 6.3 & 幅角原理及其应用     \\  \hline \hline
14  & *7 & 共形映射   & 7.1 & 解析变换的特性     \\  \hline
  &	&				 & 7.2 & 分式线性变换     \\  \hline
  &	&				 & 7.3 & 某些初等函数所构成的共形映射     \\  \hline
  &	&				 & 7.4 & 关于共形映射的黎曼存在与唯一性定理     \\  \hline \hline
   & *8 & 解析延拓   & 8.1 & 解析延拓的概念与幂级数延拓     \\  \hline
  &	&				 & 8.2 & 透弧解析延拓、对称原理     \\  \hline
  &	&				 & 8.3 & 完全解析延拓及黎曼面的概念     \\  \hline \hline
   & *9 & 调和函数  & 9.1 & 平均值定理与极值原理     \\  \hline
  &	&				 & 9.2 & 泊松积分公式与狄利克雷问题     \\  \hline \hline
15 & & & & 	{\color{red}期末考试}  \\  \hline
\end{tabular}
\end{table}

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\end{document}

复数与复变函数
解析函数
复变函数的积分
解析函数的幂级数
解析函数的洛朗展式
留数理论及其应用
共形映射
解析延拓
调和函数


